TY - CHAP

T1 - The Discrete Approach and Boundary Behavior

AU - Schuss, Zeev

N1 - Publisher Copyright:
© 2010, Springer Science+Business Media, LLC.

PY - 2010

Y1 - 2010

N2 - The path integral or the Wiener measure interpretation of stochastic differential equations is useful for both the conceptual understanding of stochastic differential equations and for deriving differential equations that govern the evolution of the pdfs of their solutions. A simple illustration of the computational usefulness of the Wiener measure is the easy derivation of the explicit expression (2.25) for the pdf of the MBM. Unfortunately, no explicit expressions exist in general for the pdf of the solution to (5.1). The second best to such an explicit expression is a (deterministic) differential equation for the pdf, whose solution can be studied both analytically and numerically directly from the differential equation. A case in point is the diffusion equation and the initial condition (2.26) that the pdf of the MBM satisfies.

AB - The path integral or the Wiener measure interpretation of stochastic differential equations is useful for both the conceptual understanding of stochastic differential equations and for deriving differential equations that govern the evolution of the pdfs of their solutions. A simple illustration of the computational usefulness of the Wiener measure is the easy derivation of the explicit expression (2.25) for the pdf of the MBM. Unfortunately, no explicit expressions exist in general for the pdf of the solution to (5.1). The second best to such an explicit expression is a (deterministic) differential equation for the pdf, whose solution can be studied both analytically and numerically directly from the differential equation. A case in point is the diffusion equation and the initial condition (2.26) that the pdf of the MBM satisfies.

KW - Absorb Boundary Condition

KW - Boundary Behavior

KW - Euler Scheme

KW - Planck Equation

KW - Stochastic Differential Equation

UR - http://www.scopus.com/inward/record.url?scp=85067982266&partnerID=8YFLogxK

U2 - 10.1007/978-1-4419-1605-1_5

DO - 10.1007/978-1-4419-1605-1_5

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AN - SCOPUS:85067982266

T3 - Applied Mathematical Sciences (Switzerland)

SP - 133

EP - 175

BT - Applied Mathematical Sciences (Switzerland)

PB - Springer

ER -